Boolean Functions: Noise Stability, Non-Interactive Correlation, and Mutual Information
Name
1801.04462.pdf
Description
Accepted version
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230.33 KB
Format
Adobe PDF
Checksum (MD5)
4f800c2c6076ba4a8374201ef929e69b
Author(s)
Li, Jiange
Medard, Muriel
Date Issued
June 2018
Publisher
IEEE
Citation
Li, Jiange and Medard, Muriel. 2018. "Boolean Functions: Noise Stability, Non-Interactive Correlation, and Mutual Information."
Version
Author's final manuscript
Abstract
© 2018 IEEE. Let T- ϵ be the noise operator acting on Boolean functions f: 0,1 nrightarrow 0,1, where ϵin[0,1/2] is the noise parameter. Given p > 1 and the mean mathbb Ef, which Boolean function f maximizes the p-th moment mathbb E(T- ϵf) p- ? Our findings are: in the low noise scenario, i.e., ϵ is small, the maximum is achieved by the lexicographical function; in the high noise scenario, i.e., ϵ is close to 1/2, the maximum is achieved by Boolean functions with the maximal degree-1 Fourier weight; and when p is an integer, the maximum is achieved by some monotone function, and in particular, among balanced Boolean functions, the maximum is achieved by any function which is 0 on all strings with fewer than n/2 1 ,s when p is large enough. Our results recover Mossel and O'Donnell's results about the problem of non-interactive correlation distillation, and confirm a conjecture of Courtade and Kumar on the most informative Boolean function in the low noise and high noise regimes. We also observe that Courtade and Kumar's conjecture is equivalent to that the dictator function maximizes mathbb E(T- ϵf) p for p close to 1.
MIT Department
Massachusetts Institute of Technology. Research Laboratory of Electronics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1109/isit.2018.8437873
